A&A 468, 1063-1073 (2007)
DOI: 10.1051/0004-6361:20066060

Fourier method of determining the rotational velocities in OB stars[*],[*]

S. Simón-Díaz1,2 - A. Herrero1,3


1 - Instituto de Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain
2 - LUTh, Observatoire de Meudon, 92195 Meudon Cedex, France
3 - Departamento de Astrofísica, Universidad de La Laguna, 38071 La Laguna, Spain

Received 18 July 2006 / Accepted 6 March 2007

Abstract
Aims. We present a comprehensive study that applies the Fourier transform to a sample of O and early B-type stars (either dwarfs, giants, or supergiants) to determine their projected rotational velocities. We then compare them with previous values obtained with other methods and seek evidence of extra broadening in the spectral lines
Methods. The Fourier technique, extensively used in the study of cooler stars, has only been marginally applied to early-type stars. The comparison of $v\sin i$ values obtained through the FT and FWHM methods shows that the FWHM technique must be used with care in the analysis of OB giants and supergiants and when it is applied to He I lines. In contrast, the FT method appears to be a powerful tool for deriving reliable projected rotational velocities and separating the effect of rotation from other broadening mechanisms present in these stars.
Results. The analysis of the sample of OB stars shows that while dwarfs and giants display a broad range of projected rotational velocities, from less than 30 up to 450 km s-1, supergiants have in general values close to or below 100 km s-1. The analysis has also definitely shown that, while the effect of extra broadening is negligible in OB dwarfs, it is clearly present in supergiants. When examining the behavior of the projected rotational velocities with the stellar parameters and across the HR diagram, we conclude, in agreement with previous researchers, that the rotational velocity should decrease when the stars evolve. On the contrary, macroturbulence may be constant, therefore resulting in an increasing importance as compared to rotation when the stars evolve.

Key words: stars: early-type - stars: rotation - techniques: spectroscopic - methods: data analysis

   
1 Introduction

Together with mass and metallicity, stellar rotation (i.e. stellar angular momentum) determines the formation, structure, and evolution of a star. Moreover, it is a necessary property for a large number of important processes, like the solar dynamo or stellar activity. In the realm of massive stars, rotation is becoming increasingly important. It is thought to be the reason for internal mixing mechanisms that would bring CNO-processed material and transport inner angular momentum to the stellar surface (Maeder & Meynet 2000). Coupled with stellar winds, it can strongly alter the evolution of the star and the theoretical Eddington limit. Equally important, the coupling with stellar winds imposes a dependence of the rotational velocity itself on metallicity during the evolution of high-mass stars (Maeder & Meynet 2001). Consequently, massive stars in the Magellanic Clouds are expected to have higher rotational velocities on average, and more CNO contamination in their atmospheres.

Since present evolutionary stellar models take this rotational velocity into account and consider it as an important parameter in the stellar evolution, accurate determination of stellar rotational velocities in massive stars is required, not only to constrain these stellar evolutionary models, but also to understand the physics of massive stars and their dependence on metallicity.

Several methods for determining of projected rotational velocity ($v\sin i$) in isolated early-type stars can be found in the literature. The most commonly used ones are

1.
The full-width-half-maximum (FWHM) of the observed line profile is measured and related to the $v\sin i$ (FWHM method).
2.
The observed spectrum is cross-correlated against a template spectrum with a low rotational velocity. The Gaussian width of the cross-correlation function is directly related to the projected rotational velocity of the star if rotation is the dominant broadening mechanism of the photospheric lines.
3.
The observed line profiles are compared with synthetic ones calculated from model atmospheres that are convolved with the corresponding broadening functions. Although this is a very accurate method if the broadening functions are well-known, too much computational time is required. Reducing that time (e.g., using model grids) also reduces the accuracy.
4.
The Fourier transform of the line profile is calculated. The position of zeroes present in this Fourier transform depends on the $v\sin i$.
Being very fast and easy to use, the FWHM method has been extensively applied to measuring the projected rotational velocities in OB stars (viz. Slettebak 1975; Herrero et al. 1992; Abt et al. 2002; Strom et al. 2005). The cross-correlation method was applied by Penny (1996), Howarth et al. (1997) and Penny et al. (2004) to IUE and STIS spectra of O-type stars and early B-type supergiants. The studies coming from both methodologies (i.e., FWHM and cross-correlation) allow us to have $v\sin i$ measurements of large samples of OB stars with different luminosity classes; however, the main limitation of these techniques is that they do not allow for the separation of rotational broadening from other possible, extra broadening mechanisms, usually referred to as macroturbulence (although the actual physical mechanism - or mechanisms - remains unknown). In fact, this is a delicate and quite important task, since many works find an absence of narrow-lined OB stars. This result is supposedly an indirect proof of a non-negligible line broadening mechanism in addition to rotation in these stellar objects, especially in the case of giants and supergiants (viz. Slettebak 1956; Conti & Ebbets 1977; Howarth et al. 1997).

The fitting to broadened synthetic profiles makes it possible to distinguish between different broadening mechanisms and also to determine other stellar parameters, but as discussed before, a large effort is required to use it accurately. Furthermore, quite good quality spectra (i.e. high signal-to-noise ratio, SNR, and spectral resolution) are required to disentangle the various broadening mechanisms that may affect the line profiles, and the derived values may depend on the considered models and the definition of the broadening functions. An early attempt to separate the rotational and macroturbulent contributions to the line broadening in O- and early B-type stars was performed by Slettebak (1956), however he found that the quality of his spectra was not good enough to get any conclusive solution.

More recently, Ryans et al. (2002) analyzed the high-quality spectra of twelve B-type supergiants. They compared observed and synthetically broadened lines using a $\chi^2$ technique and find for their survey of B-type supergiants that, while models dominated by rotation provide unsatisfactory fits, those where macroturbulence dominates and rotation is negligible (as a broadening agent) were acceptable.

Even though it is also very fast and easy to use, the Fourier method, a technique pioneered using cool stars, has been only marginally applied in the study of OB stars. This technique allows the $v\sin i$ to be easily derived independently of any other broadening mechanism affecting the line profile. There are numerous papers illustrating its use in determining $v\sin i$ in later type stars (Smith 1976; Gray 1980; Royer et al. 2002a,b; Reiners & Schmitt 2003); however, there is no comprehensive study applying this technique to early-type stars except the one by Ebbets (1979), who analyzed a sample of 16 objects, but the limited quality of the spectra allowed him to obtain estimations of the macroturbulence for only half the objects in the sample. Recently, Simón-Díaz et al. (2006) have applied the Fourier method to a small sample of OB stars in the Orion nebula. The strength of this method was shown in the study of the main ionizing star of this nebula, an O7 V star with a well-known rotational period of $\sim$15 days, which conflicts with the $v\sin i$ derived from the FWHM method, but agrees with the Fourier technique.

Following the ideas of the present paper and Ryans et al. (2002), Dufton et al. (2006a) have recently analyzed the high-resolution spectra for 24 SMC and Galactic B-type supergiants to estimate the contributions of both macroturbulence and rotation to the broadening of their metal lines. See also the analysis of periodically variable B supergiants by Lefever et al. (2006).

This paper has three main objectives. First, we extend the work presented in Simón-Díaz et al. (2006) illustrating the strength of the Fourier method in the analysis of early type stars. Second, we provide new, more accurate values for the rotational velocities of OB stars and estimate the role played by other broadening mechanisms; and third, we extend the work by Dufton et al. (2006a) and Lefever et al. (2006) towards early spectral types.

From our archive we restored the spectra of several OB stars, obtained in different observing campaigns, for a comprehensive determination of the rotational velocities through the Fourier analysis method. But first, a formal study was made to determine the range of applicability of this methodology to OB stars.

The paper is structured in four main parts as follows, beginning with a short review of the Fourier methodology, as well as some formal tests for the case of massive stars in Sect. 2. The determination of the projected rotational velocities of the sample of OB type stars is presented in Sect. 3. Finally, the discussion and the main conclusions are presented in Sects. 4 and 5.

   
2 Testing the Fourier method in early-type stars

The theoretical basis of this methodology was well-established by Gray (1976; see also the latest edition of the book, published in 2005). Here, we briefly summarize the main characteristics of the method, as needed to understand its advantages and limitations and to evaluate the results presented in Sect. 3. We also present examples specifically aimed at its application to early-type stars. For a more precise description of the methodology and its applications to late-type stars, we refer to Gray's book (see above for references). Additional details may be found in the pioneering articles by Gray (1973, 1975), Smith (1976), and Smith & Gray (1976). Some objections to its validity in special cases can be found in Mihalas (1979) and Bruning (1984).

In a simple way, an observed stellar line profile may be considered to consist of the convolution of an intrinsic profile with the rotational, macroturbulent, and instrumental profiles. The intrinsic profile includes natural and thermal broadenings, as well as other contributions, such as those due to microturbulence and Stark effect. The Fourier method ( FT) of determining $v\sin i$ (cf. Gray 1976) is based on the fact that, between the previous broadening profiles, only the rotational function is expected to have zeroes in its Fourier transform (except for high values of microturbulence; see below). As first described by Carroll (1933), the position of these zeroes in frequency space depends on the $v\sin i$ of the star, so that the frequency of the first zero ( $\sigma_{1}$) is related to the rotational velocity through

 \begin{displaymath} \frac{\lambda}{c}~v\sin i~ \sigma_{1} = 0.660. \end{displaymath} (1)

Since convolutions transform into products in Fourier space, these zeroes will also appear in the total transform function of the line profile. Therefore, it is in principle possible to determine, independently of any other broadening mechanism, the $v\sin i$ of a star once the position of the first of those zeroes is identified in the Fourier transform of the line profile[*].

Other mechanisms may contribute zeroes to the total Fourier transform, depending on their profiles in the wavelength domain. Note, for example, that microturbulence can also add zeroes, when strong lines are considered, due to the boxiness caused by it in saturated lines (Gray 1976). However, those zeroes arising from saturation in the Fourier space appear at higher frequencies than those for rotation, so there is rarely any confusion between them, except for those cases with low $v\sin i$.

   
2.1 Some formal tests for massive stars

The properties and advantages of the Fourier method, particularly in distinguishing among physical processes that produce very similar profiles in the wavelength domain, have already been discussed and illustrated in the works mentioned above. Here we present a few applications to the particular case of early-type stars.

For this formal test we used F ASTWIND (Santolaya-Rey et al. 1997; Puls et al. 2005) theoretical profiles from a model with $T_{\rm eff}$ = 30 000 K, log g = 4.0 (an early B dwarf) and convolved with different broadening profiles. Thermal and microturbulence broadenings were included in the line calculation in the usual way.

For the rotational broadenings we considered the usual formulation adopted for the rotational broadening (Gray 1976), which assumes a rotational profile that is convolved with the flux profile. As shown by Conti and Ebbets (1977), this approximation may produce a small overestimation of the derived rotational velocity of fast rotators. However, as shown by these authors, the effect is negligible below 200 km s-1 and remains weak above that value.

The treatment of macroturbulence is a more delicate issue. Several authors have pointed out indirect evidence of some type of macroturbulent broadening acting in the case of OB stars (viz. Slettebak 1956; Conti & Ebbets 1977; Howarth et al. 1997); however, its behavior is still not understood well. To a first order, we consider an isotropic macroturbulence described by a Gaussian distribution of velocities. Some notes on other possible descriptions of the macroturbulent broadening are given in Sect. 4.

Finally, in some cases, noise was added to the synthetic spectra by using a Poissonian distribution of noise generated with RANDOMN in IDL.

A first test was performed to study how well this method works when different lines from various elements present in the stellar spectrum of OB stars are considered. Our main purpose here is to see whether the Stark broadening affects the use of H and He lines as $v\sin i$ indicators. This is particularly important when comparing metal and He lines, as the latter have sometimes been proposed for $v\sin i$ measurements in early type stars using the FWHM method, particularly when the metal lines were faint and shallow, either because of the very early spectral type or the high rotational velocity.

To graphically compare results from different lines, a $\sigma \lambda $ baseline was used, as the position of the first zero associated with the $v\sin i$ depends on $\lambda $ (this is equivalent to a 1/v scale, see Eq. (1)). Figure 1 shows the result of this study. The Fourier technique was applied to several F ASTWIND synthetic lines from different elements. As can be seen, the agreement between the various lines is excellent (even when the lines He I $\lambda $4387 or H$_{\beta }$ are used), which indicates that even the strongly Stark broadened Balmer lines can be used with the Fourier method, in principle, as $v\sin i$ diagnostics. Note that the power of the Fourier transform of the He I and H lines concentrates at low frequencies, diminishing more rapidly than for the other lines when higher frequencies are considered. This is an effect produced by the presence of non-rotational broadening mechanisms affecting the line profile (i.e. Stark broadening). As in the case of lines affected by macroturbulence, it will have consequences in noisy spectra.

  \begin{figure} \par\includegraphics[angle=90,width=8.3cm,clip]{6060f1.eps}\end{figure} Figure 1: Four different synthetic line profiles generated with F ASTWIND were convolved with a $v\sin i$ of 60 km s-1. The different lines show the FT of the lines on a $\sigma \lambda $ baseline. The first zero fits perfectly to its theoretical value for all the lines. The FT can detect the $v\sin i$ feature even for the He I $\lambda $4387 and H$_{\beta }$ lines. Note that the FT of the H line reaches the value of 1 for $\sigma \lambda < 200$.
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How are these results affected by the noise present in real spectra? As mentioned by Smith & Gray (1976), random noise produces a "white noise'' in Fourier space (i.e. statistically constant with Fourier frequency). Therefore at higher frequencies, where the power of the signal transform drops, the "white noise'' dominates and hides the characteristics of the signal transform. This effect is illustrated in Fig. 2. A strong synthetic O II line was convolved with two sets of rotational and macroturbulent broadenings: ($v\sin i$, $v_{\rm m}$) = (50, 0) and (50, 50) km s-1. Additionally, a Poissonian distribution of noise, corresponding to an SNR = 200, was added to both synthetic lines. Figure 2 compares the Fourier transforms corresponding to the pure and noisy synthetic profiles. For these cases, the white noise level is located at $\sim$10-2. When the profile is only affected by a rotational broadening, the signal corresponding to the first zero is far above the noise level. This first zero can be perfectly identified, and hence the $v\sin i$ can be accurately measured. The effect of the noise is more dramatic when the line is also broadened by macroturbulence; in this case, most of the power occurs at low frequencies in the Fourier domain and the noise level is above the signal transform even for the first rotational zero. Consequently, the unambiguous identification of the first zero in the Fourier transform becomes difficult or impossible, depending on the combination of $v\sin i$, $v_{\rm m}$ and SNR.

A similar behavior will occur for the H , He I, and He II lines. Although, in principle, it is possible to use these lines to derive the $v\sin i$ (as shown above), since these lines are broadened by mechanisms other than rotation, the signatures corresponding to this rotational broadening may be hidden below the noise level due to the lowering of the low-frequency power caused by Stark broadening.

  \begin{figure} \par\includegraphics[angle=90,width=8.2cm,clip]{6060f2a.eps}\par\vspace*{2mm} \includegraphics[angle=90,width=8.2cm,clip]{6060f2b.eps}\end{figure} Figure 2: Effect of the noise on the Fourier transform of a line profile. A Poissonian distribution of noise generated with IDL and corresponding to an SNR = 200, was added to a synthetic O II line, broadened to a $v\sin i$ = 50 km s-1 ( top) or to a combination of $v\sin i$ = 50 km s-1 and $v_{\rm m}$ = 50 km s-1 ( bottom). The figure compares the transforms of the line with and without noise (dashed and solid lines, respectively).
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To obtain an estimation of the feasibility of the FT method when different quality (SNR) spectra are considered, we constructed a grid of synthetic Si III $\lambda $ 4552 lines convolved with various values of $v\sin i$ and $v_{\rm m}$ and degraded to different SNR values. The FT method was then used to determine the $v\sin i$ for the different cases (see results summarized in Table 1). Although we do not expect the noise to systematically affect the position of the zeroes, this academic example should allow us to better understand the applicability limits of the FT method in real spectra.

Table 1: Formal study of the effect of noise on the $v\sin i$ determination when a stellar line (Si III $\lambda $4452) is also affected by macroturbulence. We limit ourselves to low values because, at higher $v\sin i$, the problem is much less significant.


  \begin{figure} \par\includegraphics[angle=90,width=8.2cm,clip]{6060f3.eps}\end{figure} Figure 3: Effect of the microturbulence on the determination of the $v\sin i$ in Fourier space.
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In our formal study, we found that the effect of decreasing the SNR is to produce somewhat higher $v\sin i$ values when combined with macroturbulence. Although this effect is not very important when the macroturbulence value is low, deriving correct values even for SNR $\le$ 100, the effect may be more critical when the amount of macroturbulent broadening dominates the rotational broadening. Note that for the case of $v\sin i$ = 60 km s-1 and $v_{\rm m}$ = 80 km s-1  a SNR $\ge $ 300 is required to be affected by an error of less than $\sim$10% in determining of the $v\sin i$. Finally, Table 1 shows how critical the effect of noise can be when the $v\sin i$ is low and the amount of macroturbulence is high (f.e. $v\sin i$ = 30 km s-1 and $v_{\rm m}$ = 80 km s-1), a situation that we may expect to find in supergiants.

The final test relates to the effect of microturbulence, which may introduce additional zeroes as mentioned before. This is illustrated in Fig. 3. For this test a rotational velocity of 60 km s-1 was adopted. We see that the Fourier transform of the strong O II $\lambda $4641 line presents zeroes that are different when microturbulent velocities of 1, 9, and 20 km s-1 are introduced. With only the information from this line, it would be easy to get an incorrect value for the projected rotational velocity when using only the information from the zeroes (particularly if we only use the first one). However, we also see in the figure that we can avoid this error by using the information from weaker lines, like O II $\lambda $4956. This line is not significantly affected by the microturbulence, and thus only the zeroes due to rotation are simultaneously present in both transforms.

   
3 Applying the Fourier method to real OB stellar spectra

In the previous section it has been shown that the FT technique allows us a straightforward determination of projected rotational velocities. When dealing with real spectra, one has to recall two important properties of the FFT algorithm:
1.
The sampling of the discrete Fourier transform ( $\Delta\sigma$) depends inversely on the product of the number of points into which the line profile is sampled and the spectral dispersion: $\Delta\sigma$ = ( $N\Delta\lambda$)-1. Therefore, the better the sampling of the Fourier transform of the line profile, the more accurate the position of the zeroes associated to $v\sin i$ can be determined.
2.
The FFT of a sequence of N real numbers is symmetric and hence the maximum $\sigma$ value that is computed is $\sigma_{\rm max}$ = (N/2)( $N\Delta\lambda$)-1 = (2 $\Delta\lambda$)-1, also known as the Nyquist frequency. From formula 1 and this computational property of the FFT, it is found that the spectral dispersion ( $\Delta\lambda$) of the stellar spectrum imposes a limit in the lowest $v\sin i$ that can be derived, given by 1.320 c  $\Delta\lambda$/$\lambda $. Figure 4 illustrates the minimum $v\sin i$ that can be measured for various characteristic $\Delta\lambda$ values.

  \begin{figure} \par\includegraphics[angle=90,width=7.8cm,clip]{6060f4.eps}\end{figure} Figure 4: Minimum $v\sin i$ that can be determined at different wavelengths for various spectral dispersions as a consequence of the combination of formula (1) with the Nyquist frequency.
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Therefore, if the spectrum has enough resolution it will be possible to determine the projected rotational velocity of the star independently of any other broadening mechanism affecting the line profiles. The applicability of the FT method in real spectra is mainly limited by the spectral dispersion (for low $v\sin i$ cases) and the SNR (mainly affecting those stellar lines broadened by a non-rotational broadening mechanism, i.e. macroturbulence and/or Stark broadening).

With these ideas in mind, a sample of stellar spectra of OB stars, obtained during various observing runs, was restored from our archive to determine their projected rotational velocities by means of the FT method. The derived values were compared with those from using the FWHM method. This allowed us to test the validity of the FT method by studying those cases in which stellar lines are broadened mainly by rotation, as well as the range of applicability of the FWHM method. Likewise, the possible presence of macroturbulence could be studied.

   
3.1 The FWHM method

The FWHM method requires an unbroadened line template that is convolved with rotational profiles corresponding to various $v\sin i$ values. Several authors have considered different possibilities for the unbroadened profiles, from sharp-lined stars observed with the same configuration as the unknowns (see Slettebak 1956, who used 10 Lac; Penny 1996, who used HD 34078; and Howarth et al. 1997, who used $\tau$ Sco), to synthetic lines from stellar atmosphere code calculations (e.g. Slettebak et al. 1975, Conti & Ebbets 1977), or even Gaussian profiles with the same EW as the observed line (see Herrero et al. 1992). By determining the FWHM from a Gaussian fit to the rotationally broadened line templates, an FWHM - $v\sin i$ calibration is then generated[*]. In this study we considered the same approach as Herrero et al. (1992).

   
3.2 The observations

We have selected a sample of Galactic OB stars from observations carried out with the Isaac Newton 2.5 m and the William Herschel 4.2 m telescopes at the Observatorio de El Roque de los Muchachos during various observational campaigns between 1989 and 2003. The Intermediate Dispersion Spectrograph ( IDS) and the medium-resolution ISIS spectrograph attached to these telescopes were used. The characteristics of the different observing runs are summarized in Table 2, where the first column identifies the campaign by telescope and year, second column gives the observation date, and the different instrument configurations used in each campaign, along with their spectral dispersion, approximate resolution, and observed spectral range, are specified in columns third to sixth, respectively.

The spectra were selected to have SNR $\ge $ 200 and spectral dispersion $\le$0.4 Å/pix (which implies a minimum $v\sin i$ of $\sim$30-40 km s-1, see Fig. 4). Some spectra were selected to have a SNR slightly below 200 only for late O-type and early B-type dwarfs, as these targets are not expected to be affected by macroturbulence, and hence the effect of noise is less critical when applying the FT method to metal lines. The sample of stars ranges in spectral types from O3 to B2, including dwarfs, giants, and supergiants (see Tables 3 to 9 and Table 10).

The characteristics of the optical spectra of OB stars do not make it possible to use the same set of stellar lines to determine the projected rotational velocities for the whole sample of OB stars. For example, while in late O-type and early B-type stars many metal lines from different elements are present in the stellar spectra, along with H and He I lines, for the earliest O-type stars the spectrum is mainly dominated by H and He II lines, and only a few faint metal lines can be found. Therefore we decided to perform the study separately following the classification presented in Sects. 3.3 to 3.6.

   
3.3 Early B and late O dwarfs: an easy task

Early B and late O dwarfs have many strong metal lines, along with some He I lines, useful to infer the stellar $v\sin i$. We do not expect important broadening mechanisms other than rotation affecting the metal lines; however, the He lines are additionally broadened by the Stark broadening.

Table 3 show the $v\sin i$ derived from several He I, Si III, C II, N II, and O II lines present in the spectra of dwarf stars with spectral types between B1 and O9 and $v\sin i$ values below 100 km s-1. Results obtained through both the FT and FWHM methods are presented, where values in brackets represent FWHM results. Mean values for He I and metal lines are also presented, with errors showing the standard deviation of line values. For those cases with $v\sin i$ above 100 km s-1, the high $v\sin i$ makes metal lines to be faint and blended, and hence not suitable to measure projected rotational velocities. Except for a few metal lines, only He I lines can be used (see Table 4).

  \begin{figure} \par\includegraphics[angle=90,width=15cm,clip]{6060f5.eps}\end{figure} Figure 5: Comparison of $v\sin i$ results derived from metal and He I lines through the FT and FWHM methods. Only those cases in which $v\sin i$ could be derived through metal lines are presented. Dotted lines in the FWHM diagram represents a difference between both determinations of 10 and 20 km s-1.
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A comparison of $v\sin i$ results is shown in Figs. 5 to 7. For dwarfs, three main conclusions arise from this comparison:


  \begin{figure} \par\includegraphics[angle=90,width=7.6cm,clip]{6060f6.eps}\end{figure} Figure 6: Comparison of $v\sin i$ results derived from metal lines through the FT and FWHM methods. Dotted lines represent differences between both determinations of 20 and 40 km s-1.
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There are two cases (CygOB2-145, and $\tau$ Sco) in which the measurement of $v\sin i$ is limited by the spectral resolution. The $v\sin i$ value resulting from the FT analysis is very close to the limit imposed by the Nyquist frequency (i.e. 1.320  $c\Delta\lambda$/$\lambda $). In addition, the $v\sin i$ derived from the FWHM method is somewhat below the FT value. A larger spectral dispersion ( $\Delta\lambda$) is needed for these stars to correctly derive the $v\sin i$ through the FT method. Moreover, even with higher resolution spectra, the broadening produced by microturbulence may be comparable to the one corresponding to the rotational velocity, so that zeroes associated with microturbulence may be found at frequencies somewhat lower than $\sigma_{1}$ in the Fourier space. These stars may be good examples for studying the effects illustrated in Fig. 3.


  \begin{figure} \par\includegraphics[angle=90,width=7.8cm,clip]{6060f7.eps}\end{figure} Figure 7: Comparison of $v\sin i$ results derived from He I lines through the FT and FWHM methods for $v\sin i$ $\ge $ 180 km s-1. In this range of projected rotational velocities, no metal lines are accessible for the $v\sin i$ determination.
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Although we are not going to treat it here in detail, we would like to mention that some studies have shown that in the cases of high $v\sin i$, the effect of differential rotation may alter the position of zeroes in the Fourier transform. We refer the reader to Reiners & Royer (2004) for a detailed study of this effect in the case of A-type stars. However, note that Reiners & Royer only find evidence of differential rotation in 4 out of 78 stars of spectral types later than ours, so we do not expect this effect to be important in our sample.

   
3.4 Early B and late O supergiants: the presence of extra broadening

The logical extension of the previous analyses points towards early B and late O supergiants, since these stars also have many narrow metal lines. Results for a sample of supergiants with spectral types ranging from O9 I to B2 I are presented in Table 5, and in Figs. 5 to 7, the results from both the FT and FWHM methods are again compared.

In the case of supergiants we find:

In Sect. 2.1 it was shown that the effect of the white noise in Fourier space can be important when extra-broadening is present. If the SNR of the spectrum is not good enough, the zeroes associated to the $v\sin i$ may be hidden below the white noise and may be mistaken for artificial zeroes associated with this white noise (see Fig. 2). A low SNR may result in higher $v\sin i$ derived values.

In Fig. 8, the line Si III $\lambda $ 4552 in the ISIS@ WHT spectrum of HD 190603 (B1.5Ia+) is used to illustrate the application of the FT method to an early B-type supergiant. The spectrum has an SNR $\sim$ 570 (see Table 5). The position of the first zero in the FT of the observed line corresponds to a $v\sin i$ = 50 km s-1. The shape of the FT of a pure rotational (+ instrumental!) profile corresponding to that value does not fit the observed case correctly for low frequencies. An extra macroturbulent broadening, $v_{\rm m}$, is needed to produce a satisfactory fit in Fourier space. Note that in this case the power of the FT is below the white noise level of the observed line, thus indicating that the FT of the observed profile may be affected by the white noise, even for those frequencies where the first zero associated with the rotation is located. The actual $v\sin i$ value may be somewhat lower (and consequently, $v_{\rm m}$ higher).

  \begin{figure} \par\includegraphics[angle=90,width=7.8cm,clip]{6060f8.eps}\end{figure} Figure 8: Fourier transform of the Si III $\lambda $ 4552 line in the ISIS@ WHT spectrum of HD 190603 (B1.5Ia+). The FT of the observed line (solid black) is compared with two synthetic profiles: (dashed red) pure rotational profile with $v\sin i$ = 50 km s-1; (dashed-dotted blue) rotational + macroturbulent profile with ($v\sin i$, $v_{\rm m}$) = (50, 45) km s-1. Black arrows show the position of the zeroes associated with $v\sin i$ values of 50 and 68 km s-1, corresponding to the values derived through the FT and FWHM methods, respectively.
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3.5 Early and mid-O dwarfs and supergiants

When moving to earlier spectral types (mid and early O-type), most of the strong metal lines present in the optical spectra of early B-type stars become very faint or even do not appear anymore. A few lines associated with high ionized states of C, N, O, and Si can be found in mid O-type stars, and a few metal lines in emission are present in the spectra of the earliest spectral types. In contrast, the spectra of mid O-type stars are dominated by strong He I and He II lines. And finally, even He I lines become weak for the hottest stars. Therefore, in the optical spectra of those stars, only a few He lines are available for the $v\sin i$determination.

In previous sections, it has been shown that He I lines result in somewhat higher $v\sin i$ values than metal lines when measured with the FWHM method. In contrast, the FT method produces excellent agreement. Therefore, application of the FT method to the He I lines in O-type stars is very promising for deriving reliable $v\sin i$ values.

It is important to note that SNR imposes a strong limitation when the FT method is applied to He I lines. Since the Stark broadening mechanism has a big influence on the He I lines, high SNR spectra are necessary to be sure that white noise in the Fourier space is not affecting the identification of the first rotational zero (as shown in Sect. 2.1). The effect of noise may be even more dramatic if macroturbulence is also affecting the He I line profiles. However, note that macroturbulent broadening is expected to be greater in giants and supergiants, while Stark effect in these stars contributes less to the broadening of the He I lines (i.e. Stark broadening scales roughly as $\rho^2$). Note also that, although the spectra of early O-type stars show quite large He II lines, the effect of Stark broadening is stronger for these lines than for He I lines. Therefore, considering the comment above, the use of these lines will require spectra with a very high SNR. We did not achieve enough SNR in any of the spectra of our sample of early O-type stars, so He II could not be used.

Tables 6 and 7 summarize the line-to-line $v\sin i$ values resulting from the FWHM and FT analyses of the sample of early and mid O-type dwarfs and supergiants. The He I lines were mainly used; however, for the supergiants, we could also use some metal lines, which some of them present in the spectral range between 5000 and 6000 Å, only observed in the WHT03 campaign. A comparison of $v\sin i$ results is again shown in Figs. 5 to 7.

From the comparison of $v\sin i$ values obtained for metal lines with the FT and FWHM methods in some of the supergiants in our sample, we can conclude that macroturbulence is also present in these stars, extending the result found in B-type supergiants to the O-type supergiants. Unfortunately, the presence of macroturbulence in the early O-type dwarfs could not be investigated because there were no metal lines available for the $v\sin i$ determination.

   
3.6 OB giants

Results for the determination of projected rotational velocities in the sample of giant stars are presented in Tables 8 and 9. Similar to previous sections, $v\sin i$ measurements from both the FT and FWHM for several lines, as well as mean values for He I and metal lines, are summarized there. We mainly used He I linesfor these stars, although a few Si IV, C IV, N II and O III lines were also available in some cases.

The analysis of the sample of Giant stars points towards two curious results:

These will be discussed in more detail in the next section.

  
4 Discussion

Figure 5 shows a comparison of $v\sin i$ results obtained for the sample of stars when applying the FT and FWHM methods to metal and He I lines. This comparison was only possible for stars with $v\sin i$ values below 200 km s-1. For those cases with a projected rotational velocity above this value metal lines appeared blended or very faint, so they were not available for deriving $v\sin i$. For the studied stars, the FT method produces coherent results between metal and He I lines, indicating the possibility of accurately deriving projected rotational velocities by this method by using He I lines. This is not the case for the FWHM method, whose He I lines $v\sin i$ associated values are somewhat higher than those derived from metal lines, even for high values of $v\sin i$ (see also Fig. 7). This result is a consequence of the He I lines being affected by Stark broadening, while we are considering a Gaussian definition of the intrinsic profiles. It indicates that the assumption made by Herrero et al. (1992) that this hypothesis can be used when applying the FWHM method to He I lines for determining $v\sin i$ if this is high, is not completely correct. The effect of pressure broadening over He I may be important even for high projected rotational velocities, although for $v\sin i$ > 250 km s-1 the relative error is small. Note that, although this discrepancy should not be present if the intrinsic profiles used for the He I lines consider the Stark broadening, the derived FWHM values may require much computational effort and be model-dependent, while FT values will not.

Since the Stark effect over the He I lines scales roughly as $\rho^2$, these lines are expected to be less affected by Stark broadening in supergiants. This explains why for some of the studied supergiants, a better He I vs. metal lines agreement is found when applying the FWHM method (Fig. 5, right). However, be careful, as the agreement between He I and metal lines does not necessarily mean that the FWHM method is giving correct values for the $v\sin i$.

Differences between the $v\sin i$ values obtained by applying the FWHM and FT methods to metal lines can be interpreted as an indicator of the presence of some extra broadening, traditionally associated to large-scale motions. As indicated above, we refer to it as macroturbulence. This is illustrated in Fig. 6. The whole sample of stars whose metal lines are available for the $v\sin i$ to be measured are considered in this figure. Note that a similar comparison with He I lines (i.e. Fig. 7) does not make sense for investigating the presence of macroturbulence using the differences between the FT and FWHM methods, since these lines are also affected by other non-rotational broadenings, as shown above. Metal lines are also affected by other line broadening mechanisms, like thermal or collisional ones, but we expect them to be negligible, as shown by the agreement found in dwarf analyses. Stars have been separated according to luminosity class. While macroturbulence can be considered negligible in dwarfs, supergiants are clearly affected by this extra-broadening mechanism; 65% of the sample of supergiants shows $v\sin i$ ( FT) $\le$ 0.8 $v\sin i$ (FWHM). The case of giants is more complicated; these objects display a wide range of $v\sin i$( FT)/$v\sin i$(FWHM) ratios.

  \begin{figure} \par\includegraphics[angle=90,width=8.2cm,clip]{6060f9.eps}\end{figure} Figure 9: Distribution of projected rotational velocities (as determined by the FT method) among the various spectral types of the stars in the sample. Both $v\sin i$$_{\rm FT}$(metal) and $v\sin i$$_{\rm FT}$(He I) are used in this figure. The horizontal dashed line represents the resolution limit for R=9000.
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Supergiants rotate as a group more slowly than dwarfs do. Except for two cases among early O-type supergiants (CygOB2-8C and HD 218039), all of them have projected rotational velocities below or very close to 100 km s-1. This has to be compared with the wide range of $v\sin i$ found for dwarfs and giants, which covers from about 15 up to 450 km s-1. In Fig. 9 we see that the region of low rotational velocities is shared by all spectral types and luminosity classes, but that of high rotational velocities is dominated by giants and dwarfs of mid and late spectral types. This result has already been indicated by Slettebak (1956) and later on by other authors (viz. Conti & Ebbets 1977; Howarth et al. 1997). These authors also propose indirect indications of the possible presence of macroturbulence in OB stars, but could not directly confirm the amount of broadening that is actually due to rotation, and therefore their $v\sin i$values are affected by the contribution of macroturbulence. The FT method shows that indeed very high rotational velocities are the cause of the large line broadening in dwarfs, and allows for a quantitative estimation of the actual $v\sin i$ in supergiants (where effects of macroturbulence are important). First results for a more limited sample were presented by Ebbets (1979). Our results also support the bimodality found by Conti and Ebbets (1977) and other authors in the $v\sin i$ distribution of main sequence stars.

Although it is clear that supergiants are affected by macroturbulence, our knowledge of its behavior and the physical mechanisms that produce this extra broadening is still very poor. In Sect. 2.1 an isotropic Gaussian macroturbulence was assumed for the formal tests; however, as shown by Dufton et al. (2006a), one cannot rule out other macroturbulent velocity distributions from the comparison with high-resolution observed line profiles. Physically, a Gaussian-angle-dependent velocity distribution similar to the one proposed by Gray (1976) to model the flow of convective cells in the Sun may also make sense in the case of B supergiants. In that case, the large-scale radial-tangential movements may be produced when the stellar material flowing outwards as a stellar wind is coupled with the stellar rotation. Observational tests to investigate the exact degree of the anisotropy of the macroturbulence should be investigated in the future. In the meantime, rough estimations of the macroturbulence are also valuable, once the projected rotational velocity has been inferred. Note that the characteristic value of the macroturbulent velocity is dependent on the approach considered for the macroturbulent velocity distribution (see p.e. Dufton et al. 2006a).


  \begin{figure} \par\includegraphics[angle=90,width=8.2cm,clip]{6060f10.eps}\end{figure} Figure 10: This plot illustrates the method used to derive $\Theta _0$, a rough estimation of the macroturbulence. The various dashed curves represent cases with different values of $\Theta _0$, from 5 to 95 km s-1. An instrumental profile (R=9000) has also been considered (see text for explanation). Symbols have the same meaning as in previous figures.
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  \begin{figure} \par\includegraphics[angle=90,width=8.2cm,clip]{6060f11.eps}\end{figure} Figure 11: Distribution of the ratio $\Theta _{\rm0}$/$v\sin i$ in the sample of stars as a function of spectral types. Only those stars with metal lines available to measure $v\sin i$$_{\rm FW}$ and $v\sin i$$_{\rm FT}$ were used (see text and Fig. 10).
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The quality of our observational dataset (mainly selected to show the applicability of the FT method to OB stars) is not good enough to proceed to a fine estimation of the macroturbulence[*]. However, taking the discussion above into account, we decided to consider a rough estimation of the amount of extra-broadening, given by $\Theta _0$, where this quantity is calculated as follows. A $v\sin i$ -FWHM calibration is created by measuring the FWHM corresponding to the convolution of an instrumental profile and various values of $v\sin i$. This calibration is applied to a second set of line profiles in which an isotropic Gaussian macroturbulence (with characteristic velocity $\Theta _0$) is also included. The corresponding $v\sin i$  $_{\rm FWHM}$ is hence inferred for various $v\sin i$ - $\Theta _0$ combinations, ending with a set of lines in the $v\sin i$  $_{\rm FWHM}$ - $v\sin i$ $_{\rm FT}$ plane (see Fig. 10). Once these two quantities are measured for a given star, the $\Theta _0$ can be estimated by interpolation between these curves. We expect the quantity $\Theta _0$ calculated in this way to be roughly proportional to the macroturbulent velocity and to behave in a similar way when varying the stellar parameters.

The influence of $\Theta _0$ relative to $v\sin i$ seems to increase when moving from dwarfs to supergiants, as can be seen in Fig. 11. We decided to plot $\Theta _0$/$v\sin i$ vs. SpT, instead of $\Theta _0$, because for high values of $v\sin i$ it is more difficult to distinguish between the different values of $\Theta _0$, as illustrated by the proximity of the dashed curves when $v\sin i$ increases. We consider this ratio more meaningful.


  \begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{6060f12.eps}\end{figure} Figure 12: HR diagram where the stars in the studied sample have been located with symbol sizes proportional to their rotational ($v\sin i$$_{\rm FT}$) velocities. As in previous figures, black, blue, and red symbols represent dwarfs, giants and supergiants, respectively.
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  \begin{figure} \par\includegraphics[angle=90,width=7.6cm,clip]{6060f13.eps}\end{figure} Figure 13: Estimates of projected rotational velocities for our sample of supergiant stars. Results from the analysis of the Galactic B-type supergiants by Dufton et al. and Lefever et al. are also included.
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Except for the two giants around O7, the highest values at each spectral type are always obtained for supergiants. Only two dwarfs have $\Theta _0$/$v\sin i$ comparable to the one found in supergiants: HD 37023, which is known to be binary, and HD 168137 (O8.5 V).

Combining both results, we conclude that the role of rotation should decrease and the role of macroturbulence increase when the stars move from dwarfs to supergiants, i.e., when they evolve. This can also be seen in Figs. 12 and 13, where we represent the stars on a HR diagram, with symbol sizes proportional to their rotational and macroturbulent velocities. To place the stars in the HRD we have taken results from various authors that used state-of-the-art NLTE, line-blanketed, unified model atmospheres (Herrero et al. 2002; Repolust et al. 2004; Urbaneja 2004; Simón-Díaz 2005; Simón-Díaz et al. 2006; Crowther et al. 2006). For those stars in the sample that have not been previously analyzed, we used the SpT- $T_{\rm eff}$ and SpT-L calibrations defined by Martins et al. (2005). Note that for some of the B0.5V stars, the stellar parameters were obtained by extrapolating the calibration by Martins et al.


  \begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{6060f14.eps}\end{figure} Figure 14: Same as in Fig. 12 but for the quantity $\Theta _{\rm0}$ (i.e. rough estimation of the macroturbulence).
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We see that the symbols in Fig. 12 may have any size close to the ZAMS, but they become smaller when we depart from it. Beyond an imaginary line parallel to the ZAMS, all symbols are very small, indicating slow rotation. Although the inclination angle is unknown, it is obvious that the effect cannot be attributed to a random effect. Clearly, all supergiants have low rotational velocities which, in view of the distribution for dwarfs and giants, indicates that the stellar rotational velocities decrease when the stars evolve. As suggested by other authors (see review by Herrero & Lennon 2004), this may be related to the loss of angular momentum caused by stellar winds.

Recently Dufton et al. (2006a) and Lefever et al. (2006) have applied the FT method to two samples of Galactic B-type supergiants observed with high spectral resolution. They found higher $v\sin i$ values for the earliest spectral types. We put together the $v\sin i$ results by these authors and our results in Fig. 14. We can see that they agree very well and that our results extend those of Dufton et al. and Lefever et al. towards higher temperatures.

Additionally, these authors could use the good quality of their data to estimate the amount of macroturbulent broadening present in the stars by means of the fit of the theoretical line profiles to the observed ones. Since we limit ourselves to a crude first-order approximation as shown above, we have to be very careful when interpreting the results concerning our estimation of what we have called macroturbulence[*]. However, it is interesting to see how this estimation varies across the HR diagram. In Fig. 13 we see that the symbol sizes representing the macroturbulent velocity have a much more uniform distribution than those of the rotational velocities, with the lowest values corresponding to dwarfs close to the ZAMS. Part of this difference may be related to projection effects (that would affect large scale turbulent motions much less than rotation). Nevertheless, according to these estimations, macroturbulence does not necessarily increase significantly when the stars evolve redwards, but may remain approximately constant. However, its role in line broadening relative to rotation increases (and perhaps also in other physical mechanisms). Note that the broad scenario suggested here (rotational velocity decreasing when the stars evolve from dwarfs to supergiants, macroturbulence being roughly constant but taking a dominant role) has been deduced from the study of a very inhomogeneous sample of OB stars. To better understand the role that rotation and macroturbulence play in the evolution of massive stars, a homogeneous study of stars in clusters of different ages and metallicities should be performed (viz. Dufton et al. 2006b; Mokiem et al. 2006).

   
5 Conclusions

In the past the Fourier method of determining projected rotational velocities has been extensively used in the analysis of late-type stars, but only marginally applied to the early-type case. In this paper we show the strength of this method for its application to OB stars.

The noise and the spectral dispersion of the observed spectra are the main limitations to the Fourier method. Spectral dispersions below 0.4 Å/pix are required to measure $v\sin i$ values below 30-40 km s-1 when using the optical spectrum of OB stars. For dwarf stars, which are not very affected by non-rotational broadenings (i.e. macroturbulence), the FT method can be successfully applied even for SNR values as low as 100. The effect of noise is greater in the case of supergiants, where absorption stellar lines are broadened by other mechanisms than rotation. If the SNR of the observed spectra is not high enough and the amount of macroturbulent broadening is close to rotational broadening, the application of the FT may result in somewhat higher $v\sin i$ values than the actual ones; SNR values above 300-400 are required for these cases.

The comparison of $v\sin i$ values obtained through the FT and FWHM methods in a sample of O and early-B stars shows that the FWHM technique must be used with care when analyzing OB giants and supergiants and when it is applied to He I lines. In contrast, the FT method appears to be a powerful tool for deriving reliable projected rotational velocities and separating the effect of rotation from other broadening mechanisms present in these stars.

The analysis of the sample of OB stars, which includes dwarfs, giants and supergiants with spectral types earlier than B2, shows that while dwarfs display a broad range of projected rotational velocities, from less than 30 up to 450 km s-1, supergiants generally have values close to or below 100 km s-1. Giants also display a wide range of projected rotational velocities. The analysis has also definitely shown that, while the effect of macroturbulence in OB dwarfs is usually negligible when compared with the rotational broadening, the effect of this extra-broadening is clearly present in supergiants.

When examining the behavior of the projected rotational velocities with the stellar parameters and across the HR diagram, we conclude, in agreement with previous researchers, that the rotational velocity should decrease when the stars evolve. On the contrary, extra broadening across the HR diagram, interpreted as macroturbulence, may be constant, resulting therefore in increasing importance as compared to rotation when the stars evolve.

Acknowledgements
We thank C. Trundle for her detailed reading of the first manuscript of this work and for many fruitful discussions and comments. We also thank M.R. Villamariz, J. Puls, D. J. Lennon, and P. Dufton for numerous insightful discussions. We acknowledge the comments by the anonymous referee. This work has been partially funded by the Spanish Ministerio de Educación y Ciencia under project AYA2004-08271-C02-01. S.-D. also acknowledges the funds by the Spanish Ministerio de Educación y Ciencia under the MEC/Fullbright postdoctoral fellowship program.

References

 

  
Online Material

Table 2: Observations used for the present work.

Table 3: Projected rotational velocities ($v\sin i$ in km s-1) for the dwarf stars with spectral types between B1 and O9. (b) blended line, (r) noise important, (d) faint line, (-) line not present in the observed spectrum (either because that line does not appear for this SpT, or because it is out of the observed spectral range).

Table 4: Same as in Table 3 but for stars with $v\sin i$ values above 110 km s-1.

Table 5: Same as Table 3 but for Early-B and late-O supergiants.

Table 6: Same as Table 3 but for mid-O dwarfs.

Table 7: Same as Table 3 but for early and mid-O supergiants. a Line in emission.

Table 8: Same as Table 3 but for OB giants with $v\sin i$ values below 100 km s-1.

Table 9: Same as Table 3 but for OB giants with high $v\sin i$.

Table 10: Summary of $v\sin i$ values derived for the sample of studied stars. All $v\sin i$ values correspond to metal lines measurements except those marked with (a), where He I were used. See text for references on effective temperature and luminosity determinations.


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